Injections of Artin groups

TL;DR

This paper investigates certain Artin groups related to mapping class groups of punctured spheres, characterizing injective homomorphisms and automorphisms through topological and combinatorial methods.

Contribution

It provides a classification of injective homomorphisms between these Artin groups and describes the automorphism group of the pure braid group on multiple strands.

Findings

Injective homomorphisms are parameterized by sphere homeomorphisms and integer maps.

Every superinjective map of the curve complex is induced by a homeomorphism.

A generating set for the automorphism group of the pure braid group is given.

Abstract

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers. We also give a generating set for the automorphism group of the pure braid group on at least 4 strands. The technique, following Ivanov, is to prove that every superinjective map of the complex of curves of a sphere with at least 5 punctures is induced by a homeomorphism.